The Sample Complexity of Search over Multiple Populations

This paper studies the sample complexity of searching over multiple populations. We consider a large number of populations, each corresponding to either distribution P0 or P1. The goal of the search problem studied here is to find one population corresponding to distribution P1 with as few samples as possible. The main contribution is to quantify the number of samples needed to correctly find one such population. We consider two general approaches: non-adaptive sampling methods, which sample each population a predetermined number of times until a population following P1 is found, and adaptive sampling methods, which employ sequential sampling schemes for each population. We first derive a lower bound on the number of samples required by any sampling scheme. We then consider an adaptive procedure consisting of a series of sequential probability ratio tests, and show it comes within a constant factor of the lower bound. We give explicit expressions for this constant when samples of the populations follow Gaussian and Bernoulli distributions. An alternative adaptive scheme is discussed which does not require full knowledge of P1, and comes within a constant factor of the optimal scheme. For comparison, a lower bound on the sampling requirements of any non-adaptive scheme is presented.Comment: To appear, IEEE Transactions on Information Theor

Active Anomaly Detection in Heterogeneous Processes

An active inference problem of detecting anomalies among heterogeneous processes is considered. At each time, a subset of processes can be probed. The objective is to design a sequential probing strategy that dynamically determines which processes to observe at each time and when to terminate the search so that the expected detection time is minimized under a constraint on the probability of misclassifying any process. This problem falls into the general setting of sequential design of experiments pioneered by Chernoff in 1959, in which a randomized strategy, referred to as the Chernoff test, was proposed and shown to be asymptotically optimal as the error probability approaches zero. For the problem considered in this paper, a low-complexity deterministic test is shown to enjoy the same asymptotic optimality while offering significantly better performance in the finite regime and faster convergence to the optimal rate function, especially when the number of processes is large. The computational complexity of the proposed test is also of a significantly lower order.Comment: This work has been accepted for publication on IEEE Transactions on Information Theor

Mycobacterium ulcerans infection: control, diagnosis, and treatment.

The skin disease Buruli ulcer, caused by Mycobacterium ulcerans, is the third most common mycobacterial disease after tuberculosis and leprosy and mainly affects remote rural African communities. Although the disease is known to be linked to contaminated water, the mode of transmission is not yet understood, which makes it difficult to propose control interventions. The disease is usually detected in its later stages, when it has caused substantial damage and disability. Surgery remains the treatment of choice. Although easy and effective in the early stages of the disease, treatment requires extended excisions and long hospitalisation for the advanced forms of the disease. Currently, no antibiotic treatment has proven effective for all forms of M ulcerans infection and research into a new vaccine is urgently needed. While the scientific community works on developing non-invasive and rapid diagnostic tools, the governments of endemic countries should implement active case finding and health education strategies in their affected communities to detect the disease in its early stages. We review the diagnosis, treatment, and control of Buruli ulcer and list priorities for research and development